Negotiations with Brussels on the UK's departure from the EU are due to start on 19 June, and Mrs May is now seeking the backing of the Democratic Unionist Party to prop up her minority government.

The DUP supports Brexit - but also wants to avoid a hard border with the Republic of Ireland and to maintain as far as possible the current access to EU markets - both of which would be jeopardised if the UK leaves without a deal in place, an outcome known as a "hard Brexit".

Ex-chancellor George Osborne said the DUP's position made Ms May's "central claim" - that no deal is better than a bad deal - "undeliverable".

'European family'

And Ms Soubry - a leading figure in the Remain campaign before last year's EU referendum - told the BBC's Sunday Politics programme that Mrs May would have to listen to businesses and "wise owls" in her government who are calling for the single market to be a priority over immigration curbs.

This is not the approach adopted by the PM, who plans to withdraw from the single market and customs union and bring net migration below 100,000.

On the BBC's Andrew Marr Show, the pro-EU former deputy prime minister Lord Heseltine described Brexit as "the cancer gnawing at the Conservative Party" and urged a "period of contemplation" on the subject before negotiations begin.

The "right leader of the Conservative Party", he claimed, could appeal to German and French presidents for a deal to keep us "within the European family" while addressing immigration concerns.

But Labour's shadow chancellor John McDonnell told ITV's Peston on Sunday that full single market membership was not "on the table" and would be seen by voters as not respecting the referendum result.

'Quite clear'

Speaking on Marr, Sir Michael said the government wanted maximum access to the EU single market and an "arrangement on immigration".

He said he believed there was a majority in the Commons for such an approach.

Media playback is unsupported on your device

Media captionSir Michael Fallon: "We want a successful Brexit that is a new partnership with Europe"

"I think everybody wants to see an agreement in the end that does respect what the British people voted for last year - makes sure that our cooperation with Europe continues, our trade with Europe continues, our security cooperation with Europe continues," he said.

Leave-backing Conservative MP Dominic Raab told the Sunday Politics the country was "quite clear that they want us to make a success of Brexit".